// TODO: link to formal syntax written in BNF.#author("2017-03-23T14:43:54+09:00","default:Uedalab","Uedalab")#noattach#mathjax#norelated* Basic Syntax [#r2da58d9]- A basic hydLa program consists of definitions of constraints and declaration of constraint hierarchies.* Syntax [#r2da58d9]- The abstract syntax of HydLa is given in the following BNF.#ref(./HydLa_BNF.png, center);- A HydLa program consists of definitions of constraints and declarations of constraint hierarchies.
** Definition [#s649ac18]
- One can define constraint using the operator " ''<=>'' ".>(name of the constraint) ''<=>'' (definition of constraint) .<- In a definition, we define a named constraint or a named constraint hierarchies with arguments using the operator " ''<=>'' ".If the definition has no arguments, we can omit parentheses. INIT <=> y = 0 & y' = 0. // definition of a constraint for the initial state of a ball. FALL <=> [](y'' = -10). // definition of a constraint for falling of the ball. BOUNCE(x) <=> [](x- = 0 => x' = -x'-). // definition of a constraint for bouncing of the ball. BALL{FALL << BOUNCE}. // definition of a constraint hierarchy for the ball.- $Constraints$ allow conjunctions of constraints and implications.- The antecedents of implications are called $guards$.- " [] " denotes a temporal operator which means that the constraint always holds from the time point at which the constraint is enabled.- Each variable is denoted by a string starting with lower case($vname$).- The notation $vname′$ means the derivative of $vname$, and $vname−$ means the left-hand limit of $vname$.** Constraint [#a67e46ed]*** Conjunction [#jd43749f]>~ (constraint1) ''&'' (constraint2) ~ or~ (constraint1) ''/\'' (constraint2)<*** Always [#e8fd1135]- ''Always'' operator in temporal logic is denoted by " ''[]'' "- This operator means the constraint is effective forever from the time when this constraint is in maximal module sets.// TODO: link to maximal module set>''[]''( constraint )<*** Guard Condition [#tdd5bc86]- The "''=>''" operator means the left-hand side is the guard condition of the right-hand side.- The constraint on right-hand side is effective only when the left-hand side is entailed.>guard ''=>'' constraint<*** Derivative [#c106f801]- The "'' ' ''" operator means a derivative of the variable with respect to time.>(variable | derivative of variable)'<*** Prev [#y9d604d8]- The " - " operator means a left hand limit of a variable at each time point.>(variable | derivative of variable)'-'<*** exp [#r04a0562]- In describing equality or inequality as the constraint, not only the '(differential) , -(prev) operators, but also the arithmetic operators (+,-(minus),*,/,^(power),**(power) )can be used.- As the special values , Napier's constant "E" and Pi "Pi"(the ratio of circumference to diameter) can be used too.*** ask [#b9a84175]- In describing formula as guard condition, these (',-,arithmetic operators) , the relational operatprs (>,<,<=,>=,=,!=) and the logical operators (!(not),&(and),/\(and),|(or),\/(or)) can be used.- The evaluation result of formula is boolean value.** Hierarchy [#oc2bb594]- Meaning of a HydLa program is a set of trajectories that satisfy maximal consistent sets of candidate constraint modules sets. - Each candidate constraint set must satisfy conditions below.~-- ∀M1, ∀M2(M1<< M2 ∧ M1 ∈ MS ⇒ M2 ∈ MS) -- ∀M(¬(∃S.M << S) ⇒ M ∈ MS) *** Equality [#i272cd72]- The "," operator means that these two constraints(constraint1,constraint2) are equivalent.>constriant1 , constraint2<*** Superiority [#s5ec52f8]- The "<<" operator means right-hand side constraint(constraint4) is superior to left-hand side constraint(constraint3)- If there is contradictions between constraint3 and constraint4, constraint4 is adopted and constraint3 is excluded.>constraint3 << constraint4<** Declaration of Constraint Hierarchies [#oc2bb594]- In a declaration, we declare constraints with priorities between them.- The operator "$<<$" is a concrete notation of the operator "$\ll$" and it describes a weak composition of constraints. For example,$A << B$ means that the constraint A is weaker than B. -If we declare a constraint without "$<<$", it means that there is no priority about the constraint.- The operator "$<<$" has a higher precedence than "$,$", that is, $A, B << C$ is equivalent to $A, (B << C)$.- The unit of constraintsthat is declared with a priority is called a module or a constraint module. - Meaning of a HydLa program is a set of trajectories that satisfy maximal consistent sets of candidate constraint modules sets at each time point. - Each candidate constraint set must satisfy conditions below:~*** Precedence [#x04d90de]- The "<<" operator has a higher precedence than the "," operator.- Example:\[\forall M_1, M_2((M_1 \ll M_2 \land M_1 \in {MS} \,) \Rightarrow M_2 \in {MS}\,)\]\[\forall M (\neg \exists ( R \ll M ) \Rightarrow R \in {MS} \,)\]** List Comprehension [#eea12551]- In modeling of hybrid systems, we often come across necessity to introducemultiple similar objects.- HydLa allows a list comprehention to easily describe models with multiple objects.- A list can be defined by the operator "$:=$", like $\texttt{X := {x0..x9}.}$- There are a two types of lists: priority lists and expression lists.*** Priority List [#rb9d66d1]- The first type of lists is priority lists.-- A priority list can be denoted by an extensional notation of the following form.
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constraint1 , constraint2 , constraint3 << constraint4.$\{MP_1, MP_2, ..., MP_n\}$
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- is equal to-- It also can be denoted by an intensionally
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constraint1 , constraint2 , (constraint3 << constraint4).$\{MP | LC_1, LC_2, ..., LC_n\}$
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- not equal to-- For example, $\{\texttt{INIT(i)}\ |\ \texttt{i in \{1,2,3,4\}}\}$ is equivalent to $\{\texttt{INIT(1),INIT(2),INIT(3),INIT(4)}\}$-- If a HydLa program includes declarations of priority lists, the elements of the lists are expanded, that is, a declaration of $\{\texttt{A, B, C}\}$is equivalent to $\texttt{A, B, C}$.*** Expression List [#c1744645]- The second of list is expression lists, that is, lists of arithmetic expressions.-- We can denote an expression list in an extensional or intensional notationas well as a priority list.-- In addition, we can use range expressions in the following form.
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(constraint1 , constraint2 , constraint3) << constraint4.$\{RE .. RE\}$.
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-- $RE$ is an arithmetic expression without variables or an arithmetic expression with a variable whose name terminates with a number such as x0 and y1.*** Example of Lists [#b3820221]- An expression list $\{1*2+1..5\}$ is equivalent to $\{3,4,5\}$.- An expression list $\{\texttt{j | i in \{1,2\}, j in \{i+1..4\}}\}$ is equivalent to $\{2,3,4,3,4\}$. - An expression list $\{\texttt{x1..x3}\}$ is equivalent to $\{\texttt{x1, x2, x3}\}$.** Assert [#m5d52a83]- The condition(ask) is described in the assert sentence.- The condition(ask) must be always satisfied in the simulation of HydLa program.- In the condition(ask), the always operator("[]") can't be used. But the condition is always judged for all the time of simulation.- The condition is judged within simulating time and is not judged over simulating time.*** Other Notations [#mc029821]- The n-th element of a list $\texttt{L}$ can be accessed by $\texttt{L[n]}$.-- The index allows an arbitrary expression that results in an integer.- The size of a list $\texttt{L}$ is denoted by $\texttt{|L|}$, which can be used as a constant value in a HydLa program.- $\texttt{sum(L)}$ is a syntactic sugar of the sum of the elements in an expression list $\texttt{L}$** Tips [#jb66d45b]- Napier's constant "E" and the ratio of circumference to diameter "Pi" also can be used as constant values.- The following trigonometric functions are available.
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ASSERT{ ask }.<* Example : Bouncing Particle [#haead636]- This program is the model that a particle is falling down into floor, bounces at collision of particle and floor and repeat these movement.- The variable "y" is the position(height) of the particle.- The floor is at "y=0".>~ INIT <=> y=5 & y'=5.~ FALL <=> [](y'' = -10).~ BOUNCE <=> [](y- = 0 => y' = -4/5*y'-).~ ~ INIT, FALL << BOUNCE.<- INIT : The constraint about the beginning value of the particle.- By "INIT", the position(y) of the particle is 5 and the speed(y') of the particle is 5 at the starting time.>INIT <=> y=5 & y'=5.<- FALL : The constraint about the falling movement of the particle.- By "FALL", the acceleration(y'') of particle is -10 in that the time is over 0.>FALL <=> [](y'' = -10).<- BOUNCE : The constraint about the bouncing movement of the particle- By "BOUNCE", in that the time is over 0, if the position(y) of the particle become 0, the speed(y') of the particle becomes the value of "-4/5*y'-".- This is the bouncing movement. And the coefficient of rebound is 4/5.>BOUNCE <=> [](y- = 0 => y' = -4/5*y'-).<- The core part of HydLa program (constraint hierarchy)- This part is describing the hierarchy of the constraints(INIT,FALL,BOUNCE).- BOUNCE is superior than FALL, and INIT is equal to them.- In other words, when the particle collide with the floor, not FALL but BOUNCE is adopted as the value of the variable y.>INIT, FALL << BOUNCE.<* Advanced syntax [#k8c62e97]*** Definition using args [#j660a34c]- The argument can be used in defining constraint.- In constraint hierarchy, the constraint is called by applying arguments.>constraint_name(arg) <=> constraint_definition .<example : Define the constraint that The variables(x,y,z) in constant>~ CONST(a) <=> [](a'=0).~ ...~ CONST(x),CONST(y),CONST(z),...<example : Define the beginning values of the variables(x1,x2,x3) >~ INIT(x,xh,xv) <=> x=xh & x'=xv.~ ...~ INIT(x1,1,0),INIT(x2,2,3),INIT(x3,0,10),...<example : Define the collision constraint of the particles(x1,x2,x3) in one dimension>~ COL(p,q) <=> []( (p- = q-) => p'=q'- & q'=p'- ).~ ...~ ... << ( COL(x1,x2),COL(x1,x3),COL(x2,x3) ).<*** Subprogram of Constraint Hierarchy [#zd88597b]- The subprogram(the part of constraint hierarchy) can be used by naming program name.- The argument can be used too.- In describing the constraint hierarchy, this subprogram is called.>~ program_name { program } .~ program_name(arg) { program } .<example : Make some constraint together.>~ INITS { INIT(x1), INIT(x2), INIT(x3) }. ~ INITS, ...<example : Make the hierarchy of two constraint together>~ CONSTMOVE { CONST << MOVE }.~ CONSTMOVE, ...<example : Define bouncing program of the particles(y1,y2,y3)>~ ...~ BP(x,h,v) {INIT(x,h,v), FALL(x) << BOUNCE(x)}.~ ...~ BP(y1,1,1),BP(y2,2,3),BP(y3,3,5), ...<*** Set [#eea12551]- The set of some variables can be used.- The syntax of the set is some way.>set_name := set .<- Example of Defining>~ es := {e1, e2, e3, e4}.~ ten := { i | i in {0..9} }.~ xten := {x1..x10} .<- The calculation of sets>~ union := xten or es .~ intersection := ten and xten .<- The element of sets>~ set[i] : The i'th element of the "set" (element is start 0)~ sum(set) : The sum of the "set"~ |set| : The number of the "set"<- Examples:>~ { constarint(set[i]) | i in {1..|set|} }~ { (constarint1(set[i]) , constarint1(set[i]) ) << constraint2(set[i], set[j]) | i in {1..|set|-1} , j in {i+1..|set|} } ~ { constraint(x) | x in set }<*** Special function [#w69ae32d]- Trigonometric functions>
~ sin(x)
~ cos(x)
~ tan(x)
~ asin(x)
~ acos(x)
~ atan(x)
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- Detailed syntax and semantics can be found [http://www.ueda.info.waseda.ac.jp/~matsusho/public/dissertation.pdf here].